1. Encoding

2. Decoding

3. Uses

4. See also

5. References

6. Further reading

Elias γ code or Elias gamma code is a universal code encoding positive integers developed by Peter Elias.{{rp|197, 199}} It is used most commonly when coding integers whose upper-bound cannot be determined beforehand.

Encoding

To code a number x ≥ 1:

1. Let N = ⌊log₂ x⌋ be the highest power of 2 it contains, so 2^N ≤ x < 2^N+1.
2. Write out N zero bits, then
3. Append the binary form of x, an N+1-bit binary number.

An equivalent way to express the same process:

1. Encode N in unary; that is, as N zeroes followed by a one.
2. Append the remaining N binary digits of x to this representation of N.

To represent a number , Elias gamma (γ) uses bits.{{rp|199}}

The code begins (the implied probability distribution for the code is added for clarity):

Decoding

To decode an Elias gamma-coded integer:

1. Read and count 0s from the stream until you reach the first 1. Call this count of zeroes N.
2. Considering the one that was reached to be the first digit of the integer, with a value of 2^N, read the remaining N digits of the integer.

Uses

Gamma coding is used in applications where the largest encoded value is not known ahead of time, or to compress data in which small values are much more frequent than large values.

Gamma coding is a building block in the Elias delta code.

== Generalizations ==

Gamma coding does not code zero or negative integers.

One way of handling zero is to add 1 before coding and then subtract 1 after decoding.

Another way is to prefix each nonzero code with a 1 and then code zero as a single 0.

One way to code all integers is to set up a bijection, mapping integers (0, −1, 1, −2, 2, −3, 3, ...) to (1, 2, 3, 4, 5, 6, 7, ...) before coding. In software, this is most easily done by mapping non-negative inputs to odd outputs, and negative inputs to even outputs, so the least-significant bit becomes an inverted sign bit:

Exponential-Golomb coding generalizes the gamma code to integers with a "flatter" power-law distribution, just as Golomb coding generalizes the unary code.

It involves dividing the number by a positive divisor, commonly a power of 2, writing the gamma code for one more than the quotient, and writing out the remainder in an ordinary binary code.

See also

• Elias delta (δ) coding
• Elias omega (ω) coding
• Posit (number format)

References

Further reading

• {{cite book |author-first=Khalid |author-last=Sayood |title=Lossless Compression Handbook |publisher=Elsevier |date=2003 |chapter=Levenstein and Elias Gamma Codes |isbn=978-0-12-620861-0}}

• Anne Arundel County, Maryland
• Anne Arundel County, MD
• Anne Askew
• Anne Aubrey
• Anneau Hans-Reinhart
• Anne Ayres
• Anne Azéma
• Anne Bancroft
• Annebault
• Anne Baxter
• Anne Bayefsky
• Anne Beadell Highway
• Anne Beauchamp
• Anne Beauchamp, 15th Countess of Warwick
• Anne Beauchamp, 16th Countess of Warwick
• Anne Begg
• Anne Bishop
• Anne Blunt, 15th Baroness Wentworth
• Anne Bogart
• Anne Boleyn
• Anne Bonney
• Anne Bonny
• Anne Boyd
• Anne Bracegirdle
• Anne Bradsteet